It easily reads that the vertex amplitude \({\mathcal {A}}_{v^*}\)  is the \(\lbrace 6j\rbrace \) -symbol associated to the tetrahedron dual to \(v^*\) , and the edge amplitude \({\mathcal {A}}_{e^*}\)  is a sign given by the spins on the sides of the triangle dual to \(e^*\) ,
and the face amplitude \({\mathcal {A}}_{f^*}\)  is the dimension \(d_{j_{f^*}}\)  of the spin representation space associated to the face \(f^*\) . See \(\textit {e.g.}\)  [1]} for detailed explanation of the sign factors.
When there's no boundary, the edge amplitude term can be absorbed in the vertex amplitude [1]}, then the state-sum can be written as
\(Z[{\bf T}]= \sum _{\lbrace j_{f^*}\rbrace }\, \prod _{f^*} (-1)^{2j_{f^*}}d_{j_{f^*}}\, \prod _{v^*} (-1)^{\sum _{i=1}^6 2 j_i}\left\lbrace  \begin{array}{ccc}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6\end{array} \right\rbrace  _{v^*}\,.\) 
